4 Evolution of energy
The deformation and failure of rock are always accompanied by energy dissipation (Xie et al., 2005), and some researcher hold the view that the energy dissipation can accurately reflect the change in the mechanical properties of rocks (Ye et al., 2001). Assuming no heat exchange between the sample and external, the work done by the external load on the rock specimen (U ) is transformed into the elastic strain energy (U e) and dissipated energy (U d), and the energy equation can be expressed as
U = U e + U d
The calculation energy under triaxial cyclic loading process is shown in Fig. 6. The elastic strain energy (U e) contained axial strain energy () and radial strain energy (), as shown in Eq. (3). is the area between the unloading axial stress-strain curve and the axial strain axis, is the area the unloading confining pressure-strain curve and the radial strain axis. The dissipated energy (U d) is the difference between the input energy (U ) and elastic strain energy (U e), andU d also contained axial dissipated energy () and radial dissipated energy (), as shown in Fig. 6.
where is input axial energy, is input radial energy. is unloading axial stress, and are unloading axial and radial strain, respectively.
Figure 7 illustrates the variation of axial input energy, radial input energy, axial strain energy, radial strain energy, dissipated energy of granite specimens under triaxial cyclic loading with cycle number (σ 3 = 20MPa). It is clear that the radial input and elastic strain energy is relatively small compared with axial input energy, axial elastic strain energy and dissipated energy. Therefore, we mainly discuss the axial input energy, axial elastic strain energy and dissipated energy in this paper. Before damage threshold strength, the axial input energy, axial elastic strain energy and dissipated energy non-linearly increase with cycle number. When T = 25 and 300°C, axial input energy almost transfer as axial strain energy. However, axial strain energy and dissipated energy are near equal when T = 600°C, it indicates that damage occur in specimen once apply loading. This characteristic corresponds well with the axial plastic strain in Fig. 4. After the damage threshold strength, the difference between axial input and elastic strain energy increases, and the axial elastic strain energy reach to the peak at the peak strength. The increasing rate of dissipated energy increases with cycle number. After the peak strength, axial elastic strain released and macro-cracks formed, which result in the quick increase of dissipated energy. As the strength and elastic modulus of specimen decreases, the increasing ratio of axial input strain decreases gradually. When the loading enter to the residual strength stage, the axial elastic strain energy remains near constant, while the dissipated energy is near parallel to the axial input energy, which indicates that axial input energy almost transfer to frication energy. However, the difference between axial energy and dissipated energy increases with cycle number, and the radial input energy increases obviously when T = 600°C. It means that the radial expansion is obvious, and a part of axial input energy transfer as radial input energy.
To investigate the effect of confining pressure on energy evolution, the variation of axial input energy, axial elastic strain energy and dissipated energy of granite specimen under triaxial cyclic loading at room temperature with cycle number is illustrated in Fig. 8. It can be seen that axial input energy and dissipated energy non-linearly increases with cycle number on S type, as shown in Figs. 8a-b. With increasing confining pressure, the axial input energy and its increasing rate increase, this characteristic is more obvious between uniaxial compression and triaxial compression when σ 3 = 10 MPa. Confining pressure increases the carrying capacity of rock, and need more dissipated energy to destroy the rock, therefore when the specimen failed, the dissipated energy increase with increasing confining pressure. Confining pressure also restrain the crack initiation and propagation, therefore the variation of dissipated energy of specimen with confining is irregular before failure. The axial elastic strain energy first increases and then decreases before reaching a plateau at residual strength stage with cycle number, as shown in Fig. 8c. With increasing confining pressure, the axial elastic strain energy value, the increasing and decreasing rate increase with increasing confining pressure. It indicates the capacity to store strain energy increases with increasing confining pressure. As more strain elastic energy is stored in specimen under higher confining pressure, the specimen destroy seriously during strain elastic energy release process, which result in the increase of decreasing rate of axial elastic strain energy with increasing confining pressure. Frication force between macro-crack is larger under higher confining pressure, which results in the larger residual strength and axial elastic strain energy.
Plastic strain and dissipated energy were widely used to define the damage parameter of rock under cyclic loading (Eberhardt et al., 1999, Yang et al., 2015, Wang et al., 2018). However, the relationship between plastic strain and dissipated energy is not discussed before. Therefore, the variation of dissipated energy of granite under triaxial cyclic loading after different high temperature treatment with axial plastic strain is illustrated in Fig. 9. It can be seen that dissipated energy non-linearly increases with axial plastic strain on convex type. At initial loading stage, there is few crack in the specimen, and need more dissipated energy to initiate crack. As the axial plastic strain increase, more cracks are induced in the specimen, which make the initiation of crack easier and result in the decrease of increasing rate of dissipated energy with axial plastic strain gradually. After the macro-crack formed, energy mainly dissipate on the frictional slippage between macro-crack, which result in the linear increases of it with plastic strain. Due to more thermal cracks are induced in the specimens when T = 600°C, the non-linear characteristic is unobvious compare with that when T = 25 and 300°C.
It can be also seen that the dissipated energy increases with increasing confining pressure at same axial plastic strain, it indicates that it need more energy to initiate crack under higher confining pressure. Confining pressure also increases friction between macro-crack, which result in the increase of rising rate of dissipated energy with plastic strain. With increasing temperature, more thermal cracks are induced in the specimen, and the dissipated energy is more sensitive to confining pressure.